%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO182+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:05:47 AM UTC 2025
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO182+2 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n016.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 14:40:32 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(distinct_points_type, type, (
% 0.20/0.40 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.40 tptp_fun_Y_1: $i)).
% 0.20/0.40 tff(tptp_fun_X_2_type, type, (
% 0.20/0.40 tptp_fun_X_2: $i)).
% 0.20/0.40 tff(apart_point_and_line_type, type, (
% 0.20/0.40 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.40 tff(line_connecting_type, type, (
% 0.20/0.40 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.40 tptp_fun_Z_0: $i)).
% 0.20/0.40 tff(distinct_lines_type, type, (
% 0.20/0.40 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(4,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.40 tff(6,plain,(
% 0.20/0.40 ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(skolemize,[status(sab)],[5])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(X!2, X!2))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (~distinct_points(X!2, X!2)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))) | distinct_points(X!2, X!2)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 ((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2))) | distinct_points(X!2, X!2)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 ((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[12, 9])).
% 0.20/0.40 tff(14,assumption,(~distinct_points(Y!1, X!2)), introduced(assumption)).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 ((~(apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> (~(apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => apart_point_and_line(Z, line_connecting(Y, X))))) <=> (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(18,axiom,(~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => apart_point_and_line(Z, line_connecting(Y, X))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[19, 16])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Z, line_connecting(Y, X)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[21, 16])).
% 0.20/0.40 tff(23,plain,(
% 0.20/0.40 ~(apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[22])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 (~(apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[23, 15])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1)),
% 0.20/0.40 inference(or_elim,[status(thm)],[24])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[26])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | distinct_points(Y, Z)) <=> (distinct_points(Y, Z) | distinct_points(X, Z))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))))), rewrite((distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[29])).
% 0.20/0.40 tff(31,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart4')).
% 0.20/0.40 tff(32,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.20/0.40 tff(34,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[33])).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.20/0.40 tff(36,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 ((distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1))) <=> ((~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[37])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2))),
% 0.20/0.40 inference(transitivity,[status(thm)],[38, 36])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, X!2) | distinct_points(X!2, X!2) | (~distinct_points(X!2, Y!1)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | distinct_points(X!2, X!2) | distinct_points(Y!1, X!2)),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[41, 35, 25, 9, 14])).
% 0.20/0.40 tff(43,plain,(distinct_points(Y!1, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(44,assumption,(apart_point_and_line(X!2, line_connecting(Y!1, X!2))), introduced(assumption)).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[45])).
% 0.20/0.40 tff(47,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(Z, X) & distinct_points(Z, Y)) <=> (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))), rewrite(((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> (distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))))), rewrite((distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[50])).
% 0.20/0.40 tff(52,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con1')).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.20/0.40 tff(55,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[54])).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[56, 46])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))) | (~((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))) | (~((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(59,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))) | (~((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2)))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(X!2, line_connecting(Y!1, X!2))) | (~((~distinct_points(X!2, Y!1)) | (~distinct_points(X!2, X!2))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[60, 57, 44, 43, 13])).
% 0.20/0.41 tff(62,plain,(~apart_point_and_line(X!2, line_connecting(Y!1, X!2))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(Y!1, Y!1))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (~distinct_points(Y!1, Y!1)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[63, 7])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2))) | distinct_points(Y!1, Y!1)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 ((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.20/0.41 tff(67,assumption,(apart_point_and_line(Y!1, line_connecting(Y!1, X!2))), introduced(assumption)).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2)))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Y!1, X!2)) | (~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, X!2))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[70, 57, 67, 43, 66])).
% 0.20/0.41 tff(72,plain,(~apart_point_and_line(Y!1, line_connecting(Y!1, X!2))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(73,plain,
% 0.20/0.41 (((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))) | distinct_points(Y!1, Y!1)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(74,plain,
% 0.20/0.41 ((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[73, 64])).
% 0.20/0.41 tff(75,assumption,(apart_point_and_line(Y!1, line_connecting(X!2, Y!1))), introduced(assumption)).
% 0.20/0.41 tff(76,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(77,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(78,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.20/0.41 tff(79,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[78, 57, 25, 75, 74])).
% 0.20/0.41 tff(80,plain,(~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 (apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.41 inference(or_elim,[status(thm)],[24])).
% 0.20/0.41 tff(82,plain,
% 0.20/0.41 (~apart_point_and_line(Z!0, line_connecting(Y!1, X!2))),
% 0.20/0.41 inference(or_elim,[status(thm)],[24])).
% 0.20/0.41 tff(83,plain,
% 0.20/0.41 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[83])).
% 0.20/0.41 tff(85,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(86,plain,
% 0.20/0.41 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | apart_point_and_line(X, Z)) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))))), rewrite((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(87,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[86])).
% 0.20/0.41 tff(88,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','ceq2')).
% 0.20/0.41 tff(89,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.41 tff(90,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.20/0.41 tff(91,plain,(
% 0.20/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[90])).
% 0.20/0.41 tff(92,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.20/0.41 tff(93,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(94,plain,
% 0.20/0.41 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(95,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[94])).
% 0.20/0.41 tff(96,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[95, 93])).
% 0.20/0.41 tff(97,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(98,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Z!0, line_connecting(Y!1, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.20/0.41 tff(99,plain,
% 0.20/0.41 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[98, 92, 82, 81])).
% 0.20/0.41 tff(100,plain,
% 0.20/0.41 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(101,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[100])).
% 0.20/0.41 tff(102,plain,
% 0.20/0.41 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))))), rewrite((~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(103,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[102])).
% 0.20/0.41 tff(104,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(105,plain,
% 0.20/0.41 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(106,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[105])).
% 0.20/0.41 tff(107,axiom,(![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','cu1')).
% 0.20/0.41 tff(108,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.20/0.41 tff(109,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.20/0.41 tff(110,plain,(
% 0.20/0.41 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[109])).
% 0.20/0.41 tff(111,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.20/0.41 tff(112,plain,
% 0.20/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.20/0.41 tff(113,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(114,plain,
% 0.20/0.42 ((apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_points(Y!1, X!2)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2)))) <=> ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(115,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_points(Y!1, X!2)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[114])).
% 0.20/0.42 tff(116,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_points(Y!1, X!2)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[115, 113])).
% 0.20/0.42 tff(117,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_points(Y!1, X!2)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(118,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Y!1, X!2))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.20/0.42 tff(119,plain,
% 0.20/0.42 (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2)) | (~distinct_points(Y!1, X!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[118, 112, 99])).
% 0.20/0.42 tff(120,plain,
% 0.20/0.42 (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2)) | apart_point_and_line(Y!1, line_connecting(Y!1, X!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[119, 43])).
% 0.20/0.42 tff(121,plain,
% 0.20/0.42 (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(Y!1, X!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[120, 80, 72])).
% 0.20/0.42 tff(122,plain,
% 0.20/0.42 (apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[121, 62])).
% 0.20/0.42 tff(123,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(124,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(125,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.20/0.42 tff(126,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[125, 57, 25, 122, 11])).
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------